Tetradecagon

In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.

Regular tetradecagon
A regular tetradecagon
Type	Regular polygon
Edges and vertices	14
Schläfli symbol	{14}, t{7}
Coxeter diagram
Symmetry group	Dihedral (D14), order 2×14
Internal angle (degrees)	154+2/7°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

Regular tetradecagon

A regular tetradecagon has Schläfli symbol {14} and can be constructed as a quasiregular truncated heptagon, t{7}, which alternates two types of edges.

The area of a regular tetradecagon of side length a is given by

${\displaystyle {\begin{aligned}A&={\frac {14}{4}}a^{2}\cot {\frac {\pi }{14}}={\frac {14}{4}}a^{2}\left({\frac {{\sqrt {7}}+4{\sqrt {7}}\cos \left({{\frac {2}{3}}\arctan {\frac {\sqrt {3}}{9}}}\right)}{3}}\right)\\&\simeq 15.3345a^{2}\end{aligned}}}$

Construction

As 14 = 2 × 7, a regular tetradecagon cannot be constructed using a compass and straightedge.[1] However, it is constructible using neusis with use of the angle trisector,[2] or with a marked ruler,[3] as shown in the following two examples.

Tetradecagon with given circumcircle:
An animation (1 min 47 s) from a neusis construction with radius of circumcircle ${\displaystyle {\overline {OA}}=6}$,
according to Andrew M. Gleason,[2] based on the angle trisection by means of the Tomahawk., pause at the end of 25 s

Tetradecagon with given side length:
An animation (1 min 20 s) from a neusis construction with marked ruler, according to David Johnson Leisk (Crockett Johnson)[3] for the heptagon, pause at the end of 30 s.

The animation below gives an approximation of about 0.05° on the center angle:

Construction of an approximated regular tetradecagon

Another possible animation of an approximate construction, also possible with using straightedge and compass.

Regular tetradecagon, approximation construction as an animation (3 min 16 s), pause at the end of 25 s

Based on the unit circle r = 1 [unit of length]

• Constructed side length of the tetradecagon in GeoGebra (display max 15 decimal places) ${\displaystyle a=0.445041867912629\;[unit\;of\;length]}$
• Side length of the tetradecagon ${\displaystyle a_{target}=2\cdot \sin \left({\frac {180^{\circ }}{14}}\right)=0.445041867912629\ldots \;[unit\;of\;length]}$
• Absolute error of the constructed side length

Up to the max. displayed 15 decimal places is the absolute error ${\displaystyle F_{a}=a-a_{target}=0.0\;[unit\;of\;length]}$

• Constructed central angle of the tetradecagon in GeoGebra (display significant 13 decimal places) ${\displaystyle \mu =25.7142857142857^{\circ }}$
• Central angle of the tetradecagon ${\displaystyle \mu _{target}={\frac {360^{\circ }}{14}}=25.7142857142857\ldots ^{\circ }}$
• Absolute error of the constructed central angle

Up to the indicated significant 13 decimal places is the absolute error ${\displaystyle F_{\mu }=\mu -\mu _{target}=0^{\circ }}$

Example to illustrate the error

• At a circumscribed circle radius r = 1 billion km (the light needed for this distance about 55 minutes), the absolute error of the 1st side would be < 1 mm.

For details, see: Wikibooks: Tetradecagon, construction description (German)

Symmetry

Symmetries of a regular tetradecagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.

The regular tetradecagon has Dih₁₄ symmetry, order 28. There are 3 subgroup dihedral symmetries: Dih₇, Dih₂, and Dih₁, and 4 cyclic group symmetries: Z₁₄, Z₇, Z₂, and Z₁.

These 8 symmetries can be seen in 10 distinct symmetries on the tetradecagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[4] Full symmetry of the regular form is r28 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g14 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular tetradecagons are d14, an isogonal tetradecagon constructed by seven mirrors which can alternate long and short edges, and p14, an isotoxal tetradecagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular tetradecagon.

Dissection

14-cube projection

84 rhomb dissection

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetradecagon, m=7, and it can be divided into 21: 3 sets of 7 rhombs. This decomposition is based on a Petrie polygon projection of a 7-cube, with 21 of 672 faces. The list OEIS: A006245 defines the number of solutions as 24698, including up to 14-fold rotations and chiral forms in reflection.

Dissection into 21 rhombs

Numismatic use

The regular tetradecagon is used as the shape of some commemorative gold and silver Malaysian coins, the number of sides representing the 14 states of the Malaysian Federation.[6]

Related figures

The flag of Malaysia, featuring a fourteen-pointed star

A tetradecagram is a 14-sided star polygon, represented by symbol {14/n}. There are two regular star polygons: {14/3} and {14/5}, using the same vertices, but connecting every third or fifth points. There are also three compounds: {14/2} is reduced to 2{7} as two heptagons, while {14/4} and {14/6} are reduced to 2{7/2} and 2{7/3} as two different heptagrams, and finally {14/7} is reduced to seven digons.

A notable application of a fourteen-pointed star is in the flag of Malaysia, which incorporates a yellow {14/6} tetradecagram in the top-right corner, representing the unity of the thirteen states with the federal government.

Compounds and star polygons
n	1	2	3	4	5	6	7
Form	Regular	Compound	Star polygon	Compound	Star polygon	Compound
Image	{14/1} = {14}	{14/2} = 2{7}	{14/3}	{14/4} = 2{7/2}	{14/5}	{14/6} = 2{7/3}	{14/7} or 7{2}
Internal angle	≈154.286°	≈128.571°	≈102.857°	≈77.1429°	≈51.4286°	≈25.7143°	0°

Deeper truncations of the regular heptagon and heptagrams can produce isogonal (vertex-transitive) intermediate tetradecagram forms with equally spaced vertices and two edge lengths. Other truncations can form double covering polygons 2{p/q}, namely: t{7/6}={14/6}=2{7/3}, t{7/4}={14/4}=2{7/2}, and t{7/2}={14/2}=2{7}.[7]

Isogonal truncations of heptagon and heptagrams
Quasiregular	Isogonal			Quasiregular Double covering
t{7}={14}				{7/6}={14/6} =2{7/3}
t{7/3}={14/3}				t{7/4}={14/4} =2{7/2}
t{7/5}={14/5}				t{7/2}={14/2} =2{7}

Petrie polygons

Regular skew tetradecagons exist as Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections, including:

Petrie polygons
B7		2I2(7) (4D)
7-orthoplex	7-cube	7-7 duopyramid	7-7 duoprism
A13	D8		E8
13-simplex	511	151	421	241

References

1. Wantzel, Pierre (1837). "Recherches sur les moyens de Reconnaître si un Problème de géométrie peau se résoudre avec la règle et le compas" (PDF). Journal de Mathématiques: 366–372.
2. Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, p. 186 (Fig.1) –187" (PDF). The American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. Archived from the original (PDF) on 2016-02-02.
3. Weisstein, Eric W. "Heptagon." From MathWorld, A Wolfram Web Resource.
4. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
5. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
6. The Numismatist, Volume 96, Issues 7-12, Page 1409, American Numismatic Association, 1983.
7. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

External links

• Weisstein, Eric W. "Tetradecagon". MathWorld.